有限自动机在分形几何中的应用

对分形集的拓扑和度量性质的研究有很长的历史，包括同胚、拟对称等价、Holder参数化和Lipschitz等价等。近年来，通过图递归集方法，全不连通的自相似集的Lipschitz分类已取得了丰富的结果。但对非全不连通的自相似集，此方法不再有效。 . 有限自动机是数学和计算机科学中的重要分支，本项目引入有限自动机这个新方法，研究非全不连通的自相似集的拓扑和度量性质。我们利用邻居自动机和符号空间，通过有效词分解来构造Lipschitz映射。主要研究以下问题：(1)对平面自相似集，寻找恰当的有效词变换，使它诱导的映射是Lipschitz的。(2)在什么条件下，一个类邻居自动机和它的简化诱导的符号空间是Lipschitz等价的? (3)把有限自动机方法应用于高维自相似集和自仿集。. 本项目建立了分形几何和经典数学的新联系，拓展了分形几何研究的新方法，给符号动力系统增添了新内容。

The study of the topological and metric properties of fractal sets, including homeomorphism, quasi-symmetric equivalence, Holder parametrization, and Lipschitz equivalence, has a very long history. In recent years, using the graph-direct set technique, the Lipschitz classification of totally disconnected self-similar sets has been made considerable progression. But for non-totally disconnected self-similar sets, this method is invalid. . The finite state automaton is an important branch in mathematics as well as computer science. In this project, we introduce the finite state automaton to study topological and metric properties of self-similar sets which are not totally disconnected. We use the neighbor automaton and symbolic space to construct the Lipschitz mapping by the decomposition of special words. We intend to investigate the following problems: (1) Constructing the suitable special words transformation and bi-Lipschitz mapping between self-similar sets in the plane; (2) Searching the suitable conditions under which a finite state automaton can be simplified but the corresponding symbolic spaces are Lipschitz equivalent; (3) Applying the finite state automaton method to higher dimensional self-similar sets and self-affine fractals.. Our study will build a new link between fractal geometry and the classical subject of mathematics, explore new methods to the study of fractal geometry, and add new content to symbolic dynamic system.